Limiting behavior of random Stieltjes partial sum: adjusted method of moments estimators

We prove strong consistency for the kth-order random Stieltjes partial sums (RSPS) whenever the underlying distribution function is supported by a finite interval and derive explicit formula for its kth moment about zero. Then, we provide formulas for the mean and variance of RSPS. We examine the power distribution with density function f(x;ν,θ)=νθ(x/θ)ν-1,0 0,θ>0$f(x;\nu ,\,\theta )=\frac{\nu }{\theta }(x/\theta )^{\nu -1},\;0 0,\,\theta >0$, where θ is an unknown parameter. In this class, we observe that the expected square relative error for the adjusted method of moments estimator for θ asymptotically is one-half of the corresponding term for the maximum likelihood estimator.